1. Field of the Invention
The present invention relates generally to the field of impedance matching and more specifically to the broadband impedance matching of antennas and other frequency-dependent loads.
2. Description of the Related Art
The descriptions and examples included herein are not admitted to be prior art by virtue of their inclusion in this section.
Broadband transformers including BALUNs (BALanced to UNbalanced transformers) and UNUNs (UNbalanced-to-UNbalanced transformers) are often implemented using a transmission line design. The much-preferred design has become known as the Guanella transformer. Such a transformer consists of a set of n uniform transmission lines with characteristic impedance Z0, wavenumber xcex2, and length l, connected in parallel at one end and series at the other. The so-called common mode of the transmission lines is then choked off using any one of several methods. Thus the input impedance at the parallel-connected end of the transformer is:                               Z          in                =                                            Z              0                        n                    ⁢                      (                                                                                Z                    L                                    n                                +                                                      jZ                    0                                    ⁢                                      tan                    ⁡                                          (                                              β                        ⁢                                                  xe2x80x83                                                ⁢                        l                                            )                                                                                                                    Z                  0                                +                                                                            jZ                      L                                        n                                    ⁢                                      tan                    ⁡                                          (                                              β                        ⁢                                                  xe2x80x83                                                ⁢                        l                                            )                                                                                            )                                              (        1        )            
If the characteristic impedance of the transmission lines is chosen to be ZL/n then                               Z          in                =                              Z            L                                n            2                                              (        2        )            
for all frequencies.
This provides for a very broadband n2:1 impedance transformation. Such transformers are widely used in broadband amplifiers, fast pulse applications, and occasionally with broadband antenna systems.
This broadband constant transformation is primarily useful for matching a resistive generator to a resistive load when both generator and load resistances are constant with frequency. For example, a traditional Guanella transformer can be used to match a 50 Ohm resistive generator to a 200 Ohm resistive load. However, when matching a resistive generator to a frequency-dependent load such as an antenna, having a transformation ratio which is constant with frequency is not always advantageous. Resonant antennas exhibit frequency-dependent input impedances which cycle though alternating series and parallel type resonances with increasing frequency.
It would therefore be desirable to develop a transformer which provides a more accurate impedance match with frequency to a load having a frequency-dependent impedance.
The problems described above are addressed at least in part by a transformer combining the desirable features of the quarter-wave transformer and the Guanella transformer. This design can provide an impedance transformation ratio which varies with frequency, f, in a desirable manner.
The utility of a frequency dependent impedance transformation ratio becomes apparent by examination of the problem of obtaining maximum power transfer between a resistive source with resistance RG and a complex, frequency-dependent load with impedance ZL(f) when matching is limited to a real impedance transformation; that is, no reactance or suseptance cancellation is employed. In this case, the optimum transformed source resistance is
Ropt=|ZL|.xe2x80x83xe2x80x83(3)
Thus it is desirable to transform the source resistance to be equal to the magnitude of the complex load impedance or, alternatively, transform the complex load impedance so that its magnitude equals the source resistance. Thus, when the magnitude of the complex load impedance varies with frequency and the source impedance is a constant resistive value (as is generally the case), it is useful to have a frequency-dependent impedance transformation ratio, xcfx81, equal to the ratio of the magnitude of the complex load impedance to the generator (source) resistance.                     ρ        =                              "LeftBracketingBar"                          Z              L                        "RightBracketingBar"                                R            G                                              (        4        )            
The transformer consists of n transmission lines connected in series at one end and in parallel at the other. The transmission lines are commensurate in length and are a quarter wave long at a particular frequency, f0. The common mode of the transmission lines is choked off using one of several techniques such as coiling the transmission lines, wrapping them around a high-permeability core, threading them through high-permeability choke beads, or any of several other methods of increasing the common-mode inductance.
In general the input impedance to such a device, Zin(f), when connected to a load ZL(f) is                                           Z            in                    ⁡                      (            f            )                          =                                            Z              0                        n                    ⁢                                    (                                                                                                                  Z                        L                                            ⁡                                              (                        f                        )                                                              n                                    +                                                            jZ                      0                                        ⁢                                          tan                      ⁡                                              (                                                  β                          ⁢                                                      xe2x80x83                                                    ⁢                          l                                                )                                                                                                                                  Z                    0                                    +                                                                                                              jZ                          L                                                ⁡                                                  (                          f                          )                                                                    n                                        ⁢                                          tan                      ⁡                                              (                                                  β                          ⁢                                                      xe2x80x83                                                    ⁢                          l                                                )                                                                                                        )                        .                                              (        5        )            
At low frequencies, where the electrical length of the lines is negligible), (xcex2l less than  less than xcfx80/2),                                           Z            in                    ⁡                      (            f            )                          =                                            Z              L                        ⁡                          (              f              )                                            n            2                                              (        6        )            
and the transformer acts as a conventional Guanella transformer thus providing an n2:1 impedance transformation ratio. This impedance transformation is provided essentially independently of the characteristic impedance of the transmission lines and is maintained as long as the electrical length of the transmission lines is short.
On the other hand, when the length of the transmission lines is approximately one-quarter of a wavelength (xcex2≈xcfx80/2), the transmission lines become impedance inverters and                               Z          in                ≈                                            Z              0              2                                      Z              L                                .                                    (        7        )            
The input impedance is now independent of n and is determined entirely by Z0 and ZL.
Thus, the characteristic impedance of the lines can be chosen such that for frequencies in the vicinity of the quarter-wave frequency, the transformer acts as a quarter-wave transformer. That is, the characteristic impedance of the lines is chosen to be
Z0≈{square root over (RG|ZL+L (f0+L )|)}.xe2x80x83xe2x80x83(8)
where is f0 is the frequency at which the lines are one-quarter wavelength long. Thus, the new transformer design combines the characteristics of the Guanella transformer with those of the quarter-wave transformer to give a frequency-dependent transformation ratio. Therefore, it will be referred to as a frequency-dependent transmission line transformer.